% Thorn documentation template
\documentclass{article}
\begin{document}

\title{IDBrillData}
\author{Carsten Gundlach}
\date{6 September 1999}
\maketitle

\abstract{This thorn creates initial data for Brill wave spacetimes.
It can create both axisymmetric data (in a 3D cartesian grid), as
well as data with an angular dependency.}

\section{Purpose}

The purpose of this thorn is to create initial data for a Brill wave
spacetime.  It does so by starting from a three--metric of the form
originally considered by Brill
\begin{equation}
ds^2 = \Psi^4 \left[ e^{2q} \left( d\rho^2 + dz^2 \right)
+ \rho^2 d\phi^2 \right] =\Psi^4 \hat{ds}^{2},
\label{eqn:brillmetric}
\end{equation}
where $q$ is a free function subject to certain regularity and
fall-off conditions and $\Psi$ is a conformal factor to be solved for.

The thorn considers several different forms of the function $q$
depending on certain parameters that will be described below.
Substituting the metric above into the Hamiltonian constraint results
in an elliptic equation for the conformal factor $\Psi$ that can be
solved numerically once the function $q$ has been specified. The
initial data is also assumed to be time-symmetric, so the momentum
constraints are trivially satisfied.

% [[ DPR: The code does not use these two different initial_data
% keywords, but the axisym parameter instead.  It should probably use
% the two different initial_data values.  I have changed the doc below
% to correspond with the current code: ]]

The thorn is activated by choosing the CactusEinstein/ADMBase parameter
``initial\_data'' to be:
%in one of the following two ways:

\begin{itemize}
  
\item initial\_data = ``brilldata'': %Axisymmetric 
Brill wave initial data
%  (but calculated in a cartesian grid!).
  
%\item initial\_data = ``brilldata3D'': Brill wave initial data with an
%  angular dependency.

\end{itemize}

To choose axisymmetric data, set the parameter {\tt axisym} to true
(the default).  Note that the data is computed on a cartesian grid in
any event.  {\tt axisym = "no"} adds an angular dependence factor,
which is detailed below.

\section{Parameters for the thorn}

The thorn is controlled by the following parameters:

\begin{itemize}

\item brill\_q (INT):  Form of the function $q$ [0,1,2] (default 2):

\begin{itemize}

\item brill\_q = 0:
\[
q = a \; \frac{\rho^{2+b}}{r^2} \left( \frac{z}{\sigma_z} \right)^2
e^{-(\rho - \rho_0^2)}
\]

\item  brill\_q = 1:
\[
q = a \left( \frac{\rho}{\sigma_\rho} \right)^b \frac{1}{1 + \left[
\left( r^2 - r_0^2 \right) / \sigma_r^2 \right]^{c/2}}
\]

\item  brill\_q = 2:
\[
q = a \left( \frac{\rho}{\sigma_\rho} \right)^b e^{-\left[
\left( r^2 - r_0^2 \right) / \sigma_r^2 \right]^{c/2}}
\]

\item If one specifies 3D data (see above), the function $q$ is multiplied
by an additional factor with an angular dependency:
\[
q \rightarrow q \left[ 1 + d \frac{\rho^m}{1 + e \rho^m}
\cos^2 \left( n \phi + \phi_0 \right) \right]
\]

\end{itemize}

\item brill\_a (REAL): Amplitude (default 0.0).

\item brill\_b (REAL):  $b$ in above expressions (default 2.0).

\item brill\_c (REAL):  $c$ in above expressions (default 2.0).

\item brill\_d (REAL):  $d$ in above expressions (default 0.0).

\item brill\_e (REAL):  $e$ in above expressions (default 1.0).

\item brill\_m (REAL):  $m$ in above expressions (default 2.0).

\item brill\_n (REAL):  $n$ in above expressions (default 2.0).

\item brill\_r0 (REAL):  $r_0$ in above expressions (default 0.0).
  
\item brill\_rho0 (REAL):  $\rho_0$ in above expressions
  (default 0.0).
  
\item brill\_phi0 (REAL):  $\phi_0$ in above expressions
  (default 0.0).
  
\item brill\_sr (REAL):  $\sigma_r$ in above expressions
  (default 1.0).
  
\item brill\_srho (REAL):  $\sigma_\rho$ in above
  expressions (default 1.0).
 
\end{itemize}

The elliptic solver is controlled by the additional parameters:

\begin{itemize}
  
\item solver (KEYWORD): Elliptic solver used to solve the
  hamiltonian constraint [sor/petsc/bam] (default ``sor'').
  
\item thresh (REAL): Threshold for elliptic solver (default
  0.00001).

\end{itemize}

% Automatically created from the ccl files 
% Do not worry for now.

\include{interface}
\include{param}
\include{schedule}

\end{document}